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I'm exploring the divergence theorem and Green's theorem, but I seem to be lacking some understanding. I have tried this problem several times, and I am wondering where my mistake is in this method.
The problem:
For one example, I am trying to find the divergence of some vector field from a hemisphere. Let the hemisphere be given by $$x^2 + y^2 + z^2 = 9.$$ Also, the vector field in question is given by $$ \textbf{V} = \bigg(y,\hspace{2mm} xz,\hspace{2mm} 2z-1\bigg) $$
Now, I want to evaluate the integral over the surface:
$$\iint\textbf{V}\cdot\textbf{n}\hspace{2mm}d\sigma$$
Attempt at a solution:
Here is how I try to solve it. I instead use (by Green's theorem, where tau is a volume element) $$\iiint\nabla\cdot\textbf{V}\hspace{2mm}d\tau.$$
Taking the gradient of the the vector field, I get 2 (only the z-hat component of the field will contribute). And since it is a simple hemisphere, I can integrate over the volume in spherical coordinates with the following limits:
$$r \hspace{1mm}\epsilon\hspace{1mm}[0,3]$$
$$\phi \hspace{1mm}\epsilon\hspace{1mm}[0,2\pi]$$
$$\theta \hspace{1mm}\epsilon\hspace{1mm}[0,\pi/2]$$
The Jacobian is standard for going from Cartesian to spherical coordinates: $$r^2 \hspace{1mm}sin(\theta)$$
Lastly, evaluating this integral (and not forgetting to include the gradient of the vector field in the integral), I get $$36\pi$$
The answer given in the text is $$27\pi$$ This is not a hard problem, and I am most certain that my integration and arithmetic is correct. There must be some fundamental step that I am missing.
The problem:
For one example, I am trying to find the divergence of some vector field from a hemisphere. Let the hemisphere be given by $$x^2 + y^2 + z^2 = 9.$$ Also, the vector field in question is given by $$ \textbf{V} = \bigg(y,\hspace{2mm} xz,\hspace{2mm} 2z-1\bigg) $$
Now, I want to evaluate the integral over the surface:
$$\iint\textbf{V}\cdot\textbf{n}\hspace{2mm}d\sigma$$
Attempt at a solution:
Here is how I try to solve it. I instead use (by Green's theorem, where tau is a volume element) $$\iiint\nabla\cdot\textbf{V}\hspace{2mm}d\tau.$$
Taking the gradient of the the vector field, I get 2 (only the z-hat component of the field will contribute). And since it is a simple hemisphere, I can integrate over the volume in spherical coordinates with the following limits:
$$r \hspace{1mm}\epsilon\hspace{1mm}[0,3]$$
$$\phi \hspace{1mm}\epsilon\hspace{1mm}[0,2\pi]$$
$$\theta \hspace{1mm}\epsilon\hspace{1mm}[0,\pi/2]$$
The Jacobian is standard for going from Cartesian to spherical coordinates: $$r^2 \hspace{1mm}sin(\theta)$$
Lastly, evaluating this integral (and not forgetting to include the gradient of the vector field in the integral), I get $$36\pi$$
The answer given in the text is $$27\pi$$ This is not a hard problem, and I am most certain that my integration and arithmetic is correct. There must be some fundamental step that I am missing.
